
We note that the norm constraint here has dimension \(dc\)  and \(\mathbf {w}_1\)  has dimension \(d^2\) , so by [1]}, [2]} this strong duality result from Lemma REF  requires that \(m^* \le n\min \lbrace d^2, dc\rbrace \) . Now, using the concept of dual norm, this is equal to
\(p_{SA}^* &= \max _{\mathbf {V}_i} -\sum _{i=1}^n \mathcal {L}^*\left(\mathbf {V}_i, \mathbf {Y}_i\right) \nonumber \\\mathrm {s.t.}~&\max _{\begin{array}{c}\Vert \mathbf {g}\Vert _2 \le 1 \\ \Vert \mathbf {w}_1\Vert _2\le 1 \\ j \in [P] \\ \mathbf {K}_j \mathbf {w}_1 \ge 0\end{array}} \mathbf {g}^\top \sum _{i=1}^N ({\mathbf {X}}_i^\top \otimes \mathbf {V}_i^\top ) \mathbf {D}_{j}^{(i)}({\mathbf {X}}_i \otimes {\mathbf {X}}_i) \mathbf {w}_1\) 
